Design of neuro-swarming computational solver for the fractional Bagley–Torvik mathematical model

This study is to introduce a novel design and implementation of a neuro-swarming computational numerical procedure for numerical treatment of the fractional Bagley–Torvik mathematical model (FBTMM). The optimization procedures based on the global search with particle swarm optimization (PSO) and local search via active-set approach (ASA), while Mayer wavelet kernel-based activation function used in neural network (MWNNs) modeling, i.e., MWNN-PSOASA, to solve the FBTMM. The efficiency of the proposed stochastic solver MWNN-GAASA is utilized to solve three different variants based on the fractional order of the FBTMM. For the meticulousness of the stochastic solver MWNN-PSOASA, the obtained and exact solutions are compared for each variant of the FBTMM with reasonable accuracy. For the reliability of the stochastic solver MWNN-PSOASA, the statistical investigations are provided based on the stability, robustness, accuracy and convergence metrics.


Introduction
The fractional Bagley-Torvik mathematical model (FBTMM) has achieved the huge attention of the research community in recent years. The fractional kinds of derivatives represent the physical network dynamics, a rigid plate based on the Newtonian fluid, and the frequency-dependent systems of the damping properties [1][2][3][4]. The numerical, approximate and analytical form of the FBTMM has been performed by many scientists and reported in [6][7][8][9][10]. While few other utmost deterministic and stochastic numerical schemes [11][12][13][14][15][16][17][18][19] are listed in Table 1 in terms of novel methodology exploited for the solutions, publication year, and necessary remarks to highlight their significance in the reported literature for FBTMM.
The present study is to solve the FBTMM by using a competent soft computing approach based on a Mayer wavelet neural network (MWNN) using the optimization procedures of particle swarm optimization (PSO) along with active-set algorithm (ASA), i.e., MWNN-PSOASA. The general form of the FBTMM is provided as [20][21][22][23]: where a α indicates the initial conditions, λ represents the derivative based on the fractional-order with 1.25, 1.5, and 1.75, v(τ ) is the solution of above Eq. (1), while a 1 , a 2 , and a 3 are the constant values The FBTMM represented in Eq. (1) was the pioneering work of Bagley and Torvik introducing on the motion of an absorbed plate using the Newtonian fluid [3]. a emails: juan.garcia@upct.es; jlgarcia@kau.edu.sa (corresponding author) b e-mail: zulqurnain_maths@hu.edu.pk c e-mail: rajamaz@yuntech.edu.tw d e-mail: dumitru@cankaya.edu.tr  [11] in 1998 Podlubny's consecutive approximation Novel numerical solution [12] in 2002 Deterministic numerical scheme Convergence established [13] in 2007 Differential transform method Novel numerical solver [14] in 2008 Adomian decomposition method Novel analytical solution [15] in 2008 He's variational iteration method Viable analytic method [16] in 2009 Matrix approach of discretization Novel discretization [17] in 2010 Shooting collocation approach Efficient scheme [18] in 2010 Taylor collocation method Power series approach [19] in 2011 Genetic algorithms and neural networks Novel stochastic solver [20] in 2011 Neural networks and Swarm intelligence Viable stochastic solver [20] in 2012 Haar wavelets operational matrix Novel wavelets approach [22] in 2017 Sequential quadratic programing Fractional neural network [23] in 2020 Interior-point method Fluid dynamics problem [24] in 2020 Galerkin approximations Numerical scheme [25] in 2020 Exponential spline approximation Novel spline method [26] in 2020 Jacobi collocation methods Power series approach [27] in 2021 Generalized Bessel polynomial Power series method [28] in 2021 Quadratic finite element mentod Numerical computing [29] in 2021 Lie symmetry analysis method Numerical analysis

Problem statement
The stochastic computing solvers have been generally applied to the singular, nonlinear and dynamical systems based on the platform of neural network together with the swarming/ evolutionary optimization schemes [30][31][32]. The stochastic solvers have been applied in diverse applications, few of them are coronavirus SITR model [33,34], singular doubly differential systems [35], fluid dynamics problems [36], HIV infections modeling systems [37,38], and electric circuits model [39,40]. The authors are motivated by keeping these stochastic-based applications to design a computing solver for the FBTMM. Therefore, the objective of study is to introduce a novel design and implementation of a neuro-swarming computational numerical procedure MWNN-PSOASA for numerical treatment of the fractional Bagley-Torvik mathematical model (FBTMM) by exploiting global search optimization procedures via particle swarm optimization (PSO) and local search via active-set approach (ASA), while Mayer wavelet kernel-based activation function used in neural network (MWNNs) modeling.

Novelty and inspiration
The novelty and significance of the research investigations are briefly described in this section. The literature review presented for FBTMM, one can decipher evidently that a large variant of deterministic solvers have been introduced by research community for solving the FBTMM while few studies of stochastic solvers are available for finding the approximate solutions for FBTMM, Therefore a novel fractional neural network is presented for the solution of FBTMM by exploiting the strength of fractional Mayer wavelet neural networks (MWNN) based modeling of the fractional derivative terms in ODE (1) and training of these networks are performed by hybrid heuristics having global search with PSO and ASA based local refinements, i.e., MWNN-PSOASA. The designed solver MWNN-PSOASA is used efficiently and effectively to solve the FBTMM numerically. The achieved form of the numerical results is compared with the accessible true/exact solutions, which shows the precision, reliability, constancy, and convergence of the designed solver MWNN-PSOASA. The reliability of the results based on the designed solver MWNN-PSOASA is further presented using the statistical procedures of Mean, semi interquartile range (S.I.R), Minimum (Min), standard deviation (STD), Theil's inequality coefficient (TIC), mean square error (MSE) and Maximum (Max). Besides, the accurate and reasonably stable outcomes of the FBTMM through the designed solver MWNN-PSOASA, robustness, smooth processes, and exhaustive pertinence are other significant perks of the scheme.

Organization
The organization of this study is considered as: The methodology of MWNN-PSOASA is accessible in Sect. 2. The performance operators are shown in Sect. 3. The comprehensive results detail is given in Section 4. Final remarks and upcoming research directions are given in the last section.

Methodology
This section represents the methodology of the designed solver by using the Mayer wavelet neural network along with the optimization of PSOASA to solve the FBTMM. The genetic flow diagram of proposed MWNN-PSOASA for solving FBTMM is provided in Fig. 1, in which the process blocks in four steps are presented. The construction of the FBTMM, merit function based on the mean square error, and the PSOASA optimization is also presented in this section.

Objective function: MWNN
In this section, the FBTMM solutions are signified byv(τ ), whereas, D (n)v (τ ) and D αv (τ ) provides the integer derivatives of order n and fractional form of the derivative. The mathematical formulations of these systems by means of continuous mapping in neural networks models are given as:v An objective function-based Mayer wavelet is written as: The updated Eq. (2) using the above values is become as: The combination of the MWNN with the optimization of PSOASA is used to solve the FBTMM based on the availability of appropriate W values. For the ANN weights, an objective function e F is given as: Here e F−1 and e F−2 are the objective functions based on the FBTMM and the ICs of Eq. (1), respectively written as:

Networks optimization: PSOASA
The parameter optimization, i.e., weights, for the MWNN models are obtained using the hybridization of computing procedures of particle swarm intelligence PSO as an efficacious global search aided with active set algorithms (ASAs) for efficient local refinement mechanism to solve the variants of FBTMM in equation (1). Particle swarm optimization is a computational swarm intelligence approach, which is used to optimize a model through the process of iteration to improve the applicant outcomes, i.e., candidate solutions of a specific optimization tasks, with respect to assume quality measures and constraints. The PSO normally solves a model by using the population of applicant outcomes called swarm and each candidate solution is represented by the particles. The PSO algorithms operate with the adjustment of these particles during each flight in search-space based on the mathematical representations of the particles velocity and position in terms of previous velocity, inertia weight of velocity, cognitive learning block via local best particle, and social learning mechanism via global search particle. The movement of the particles is affected by its local prominent based positions; however, it is also directed to the best-recognized positions, which are efficient as improved positions of other particles. This is projected to transfer the swarm to the best results. Additional necessary elaborative details, underlying theory, mathematical representation, scope, and applications in diversified fields can be seen in [41][42][43] and references mentioned in them. In recent decades, PSO is implemented to plant diseases diagnosis and prediction [44], nonlinear Bratu systems governing the fuel ignition model [45], identification of control autoregressive moving average systems [46], reactive power planning [47], and thermal cloaking and shielding devices [48].
In order to control and speed up the convergence performance of the global search PSO, the optimization through the hybridization with local search method is implemented for speedy adjustment of the parameters. The active-set algorithm is one of the quick, rapid, and efficient local search schemes, which is famous to find the optimal performances in different fields. ASA is an effectual convex optimization tool that is implemented for unconstrained and constrained systems. Few prominent applications of utmost performance of the ASA include embedded system of predictive control [49], pressure-dependent network of water supply systems [50], local decay of residuals in dual gradient method [51], nonlinear singular heat conduction model [52], and warehouse location problem [53]. Therefore, in the presented study, a memetic computing paradigm PSOASA based on global search efficacy of PSO aided with speedy tuning of parameter with ASA are exploited for finding the known adjustable of MWNN models for solving the FBTMM in Eq. (1).

Results and discussions
In this section, the solutions of three different variants based on the FBTMM are provided by using the integrated design heuristics of MWNN-PSOASA. The precision and convergence on the basis of sixty number of autonomous trails MWNN-PSOASA are presented using sufficient large number of graphical and numerical illustrations with elaborative details for solving the variants of the FBTMM. While the information/outcomes of different performance indices for the analysis of proposed MWNN-PSOASA are given in this section.
The mathematical form of the performance gages of TIC and MSE are presented to solve the FBTMM as follows: where n is total number of grid points, i.e., τ m , m 1, 2, …, n, while v m is the reference solution for m th grid point whilev m is proposed approximate solution for the m th grid point.
Comparison of the outcomes of fractional MWNNs optimized with PSOASA for solving FBTMM has been made with reported results of state of the art deterministic and stochastic solver in order to access the performance rigorously. The absolute error of the reported numerical solver based on matric approach introduced by Podlubny [11], sigmoidal fractional neural networks optimized with IPA (FNN-IPA) [22,23], sigmoidal neural networks optimized with GAs aided with pattern search (PS), i.e., GA-PS [54] and sigmoidal neural networks trained with particle swarm optimization (PSO) supported with PS, i.e., (PSO-PS) [20] are presented in Table 4 along with the proposed results of FMWNN-PSOASA. One can easily decipher from results presented in Table 4, the values of the AE for FMWNN-PSOASA are comparable to state-of-the-art deterministic and stochastic numerical procedures for solving FBTMM.

Concluding remarks
The current work investigations are to design a neuro-swarming computational numerical procedure for the fractional Bagley-Torvik mathematical model. The optimization procedures based on the global search particle swarm optimization and local search active-set approach using the activation function Mayer wavelet neural network have been applied to solve the fractional model. The proposed stochastic solver MWNN-GAASA efficiency is performed to solve three different variants based on the fractional order of the FBTMM. For the exactness of the stochastic solver MWNN-PSOASA, the comparison of the attained and exact solutions will be provided for each variant of the FFBTMMM. The AE values have been obtained in good measures that are calculated around 10 −06 to 10 −07 for each example of the FBTMM. For the reliability of the proposed stochastic solver MWNN-PSOASA, the statistical soundings are provided based on the stability, robustness, accuracy, and convergence. One can conclude from these outcomes that around 75% independent trials achieved precise level of the accuracy. Beside the advantage of accurate and reliable outcomes of designed MWNN-PSOASA, the limitation of slowness of operation of global search with PSO and then local search with ASA.

Further research openings
The FMWNN-PSOASA can be implemented to solve the fluid nonlinear models, fraction order systems, and fluid models [55][56][57][58][59][60][61][62][63]. Moreover, the used of heuristic methodologies having inherent strength of global as well as local search like differential evolution, backtracking search optimization algorithm, weights differential evolution, and their recently introduced variants are good alternative of integrated PSOASA.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This paper has been partially supported by Fundación Séneca de la Región de Murcia grant numbers 20783/PI/18, and Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-0971-B-100.

Data Availability Statement
There is no data associated with this manuscript.

Declarations
Conflict of interest All the authors of the manuscript declared that the authors have no conflicts to disclose.
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